Smooth diffeomorphisms with countable set of stable periodic points are presented. A neighborhood of a homoclinic point may contain infinitely many stable periodic points, but at least one characteristic exponent of such points tends to zero with increasing the period. By Rolle's theorem, the second derivative of g vanishes at the points. Similarly, it is easy to see that the derivative of any order higher than the second of the function g vanishes at infinitely many points in any neighborhood of zero. Conditions were obtained under which any neighborhood of a homoclinic point of a diffeomorphism contains infinitely many stable periodic points whose characteristic exponents are bounded away from zero.
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- Математика (все)